Ion Implantation and Synthesis of Materials - Studium

343 Dynamics of Binary Elastic CollisionsUsing (3.25) to define the orbital angular momentum, we rewrite (3.26) in theformV( rmin) b0 = 1− − ,E r22min(3.27)where E = E c in the CM system. Equation (3.26) shows that r min will depend onthe energy of the ion and the form of the interatomic potential. Knowledge of V(r)allows one to find r min by applying the quadratic equation to (3.27).3.6 Distance of Closest ApproachThe distance of closest approach was defined by (3.27) and can be rewritten asV( r )Mmin 2= E2 2 c= E0/min 1 21−b r M + M.(3.28)For a Coulomb potential and a head-on collision, b = 0, we can rewrite (3.28) asM + M Z Z ed r Z Z e1 2 2 1 2c≡ (min)=1 2=b = 0M2E0 Ec2(3.29)for a head-on collision where d c is called the collision diameter, r min ≡ d c . For agiven interatomic potential and ion energy, the collision diameter gives the lowerlimit to r min .As an example, for 1 MeV He (Z 1 = 2) ions incident on Si (Z 2 = 14), the CMenergy, E c = M 2 E 0 /(M 1 + M 2 ), equals 875 keV and the distance of closest approach(the collision diameter) equals Z 1 Z 2 e 2 /E c = 4.6 × 10 −5 nm, which has a value muchsmaller than a TF . It is informative to write d c relative to the screening distance, a TF ,where the parameter a TF /d c is referred to as ε, the reduced energy, given bya a E E a Mε ≡ = =d Z Z e Z Z e M MTF TF cTF 22 2c 1 2 1 2 1+2.(3.30)Examining (3.30) we see that ε is a dimensionless energy unit. Physically, ε givesa measure of how energetic the collision is and how close the ion gets to thenucleus of the target atom. For example, the value of the Thomas-Fermi screeningdistance, a TF , for He on Si isa0.885×0.053−2= = 1.5×10 nm.1/2 1/2( Z1 + Z2)TF 2/3